Problem: What is the value of $\dfrac{d}{dx}(x^4-4x^2-10x)$ at $x=3$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $5$ (Choice B) B $15$ (Choice C) C $92$ (Choice D) D $74$
Answer: Let's first find the expression for $\dfrac{d}{dx}(x^4-4x^2-10x)$ and then evaluate it at $x=3$. According to the sum rule, the derivative of $x^4-4x^2-10x$ is the sum of the derivatives of $x^4$, $-4x^2$, and $-10x$. The derivatives of these terms can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ For example, this is the derivative of the first term: $\dfrac{d}{dx}(x^4)=4x^3$ Here is the complete differentiation process: $\begin{aligned} &\phantom{=}\dfrac{d}{dx}(x^4-4x^2-10x) \\\\ &=\dfrac{d}{dx}(x^4)-4\dfrac{d}{dx}(x^2)-10\dfrac{d}{dx}(x)&&\gray{\text{Basic differentiation rules}} \\\\ &= 4x^3-4\cdot 2x-10\cdot 1x^0&&\gray{\text{The power rule}} \\\\ &=4x^3-8x-10 \end{aligned}$ So we found that $\dfrac{d}{dx}(x^4-4x^2-10x)=4x^3-8x-10$. Plugging in $x=3$ and evaluating using the calculator, we find that the value is $74$. In conclusion, the value of $\dfrac{d}{dx}(x^4-4x^2-10x)$ at $x=3$ is $74$.